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Light-by-light scattering amplitudes from generalized unitarity in massive QED

Christophe Bernicot

TL;DR

This paper develops and applies generalized unitarity to compute all one-loop four-photon helicity amplitudes in massive QED, covering scalar QED, spinor QED, and ${ m QED}^{ exists=1}$ SUSY. By employing two-, three-, and four-cut techniques and a master-integral basis including extra-dimension integrals, it isolates and reconstructs the coefficients of $I_4^{n+2}$, $I_3^{n}$, and $I_2^{n}$, and identifies the origin of rational terms with $oldsymbol{bc}$-dependent contributions. The work delivers compact, cross-checked formulas for all helicity configurations, and uses a supersymmetric decomposition to relate scalar, fermion, and ${ m N}=1$ amplitudes, including a complete analytic result for the all-plus six-photon amplitude. Together, these results validate the efficiency of unitarity-based methods in massive theories and provide a foundation for more complex multi-leg processes relevant to high-energy phenomenology.

Abstract

We calculate all the four-photon helicity amplitudes at the one-loop level in a massive theory using multiple-cut methods. The amplitudes are derived in scalar QED, QED and $\textrm{QED}^{\caln =1}$ theories. We will see the origin of rational terms. We extend the calculation to the simplest six-photon helicity amplitude where all photons have the same helicity.

Light-by-light scattering amplitudes from generalized unitarity in massive QED

TL;DR

This paper develops and applies generalized unitarity to compute all one-loop four-photon helicity amplitudes in massive QED, covering scalar QED, spinor QED, and SUSY. By employing two-, three-, and four-cut techniques and a master-integral basis including extra-dimension integrals, it isolates and reconstructs the coefficients of , , and , and identifies the origin of rational terms with -dependent contributions. The work delivers compact, cross-checked formulas for all helicity configurations, and uses a supersymmetric decomposition to relate scalar, fermion, and amplitudes, including a complete analytic result for the all-plus six-photon amplitude. Together, these results validate the efficiency of unitarity-based methods in massive theories and provide a foundation for more complex multi-leg processes relevant to high-energy phenomenology.

Abstract

We calculate all the four-photon helicity amplitudes at the one-loop level in a massive theory using multiple-cut methods. The amplitudes are derived in scalar QED, QED and theories. We will see the origin of rational terms. We extend the calculation to the simplest six-photon helicity amplitude where all photons have the same helicity.

Paper Structure

This paper contains 47 sections, 155 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and explanations
  3. The structure of the amplitude $\gamma_{1}+\gamma_{2}+\gamma_{3}+ \gamma_{4} \rightarrow 0$
  4. Spinor formalism and helicity amplitudes
  5. The regularization scheme and notation of integrals
  6. Generalized unitarity-cuts
  7. $A_{4}^{scalar}$ helicity amplitudes
  8. $A_{4}^{scalar}(1^{+},2^{+},3^{+},4^{+})$ helicity amplitude
  9. Four-cut technique
  10. Three-cut technique
  11. Two-cut technique
  12. $A_{4}^{scalar}(1^{-},2^{+},3^{+},4^{+})$ helicity amplitude
  13. Four-cut technique
  14. Three-cut technique
  15. Two-cut technique
  16. ...and 32 more sections

Figures (3)

  • Figure 1: Fermion loop with a cut in the channel $s_{12}$.
  • Figure 2: Kinematics of bubbles, one external mass triangle and no external mass box.
  • Figure 3: Chain composed with two photons, with two different helicity, following with one photon.